 Remark 2.1 For the continuity of \(\mathcal { L}\)  in \(C^0(\overline{\Omega })\) , we can use the well-known De Giorgi-Nash estimate: \(\lbrace w_i\rbrace \)  is bounded in \(C^{0,\alpha } (\overline{\Omega })\)  if
\(u_b\in C^{0,\alpha _0} ( \partial \Omega )\)  and \(\Omega \)  satifies a uniform exterior cone condition (Theorem 8.29 [1]}).
